Related papers: The net-regular strongly regular signed graphs wit…

In this paper, an upper bound on the nullity of signed graphs in terms of the cyclomatic number and the number of pendant vertices is proved, and the corresponding extremal signed graphs are completely characterized.

Let $\mathscr{L}_{n,t}$ be the set of all $n$-vertex connected graphs with clique number $t$\,($2\leq t\leq n)$. For $n$-vertex connected graphs with given clique number, lexicographic ordering by spectral moments ($S$-order) is discussed…

The purpose of this paper is to characterize graphs that do not have a large $K_{2,n}$-minor. As corollaries, it is proved that, for any given positive integer $n$, every sufficiently large 3-connected graph with minimum degree at least…

In this paper we determine the exact values of the signed domination number, signed total domination number, and minus domination number of complete multipartite graphs, which substantially generalizes some previous results obtained for…

We introduce a notion of genus range as a set of values of genera over all surfaces into which a graph is embedded cellularly, and we study the genus ranges of a special family of four-regular graphs with rigid vertices that has been used…

We say that a signed graph is $k$-critical if it is not $k$-colorable but every one of its proper subgraphs is $k$-colorable. Using the definition of colorability due to Naserasr, Wang, and Zhu that extends the notion of circular…

In this survey we overview known results on the strong subgraph $k$-connectivity and strong subgraph $k$-arc-connectivity of digraphs. After an introductory section, the paper is divided into four sections: basic results, algorithms and…

A strong edge-coloring of a graph $G$ is an assignment of colors to edges such that every color class induces a matching. We here focus on bipartite graphs whose one part is of maximum degree at most $3$ and the other part is of maximum…

A graph has strong convex dimension $2$, if it admits a straight-line drawing in the plane such that its vertices are in convex position and the midpoints of its edges are also in convex position. Halman, Onn, and Rothblum conjectured that…

A signed graph is a graph whose edges are labeled either positive or negative. Corresponding to the two signed distance matrices defined for signed graphs, we define two signed distance laplacian matrices. We characterize balance in signed…

We consider intrinsic linking and knotting in the context of directed graphs. We construct an example of a directed graph that contains a consistently oriented knotted cycle in every embedding. We also construct examples of intrinsically…

In 1972 Tutte famously conjectured that every 4-edge-connected graph has a nowhere zero 3-flow; this is known to be equivalent to every 5-regular, 4-edge-connected graph having an edge orientation in which every in-degree is either 1 or 4.…

The definition of edge-regularity in graphs is a relaxation of the definition of strong regularity, so strongly regular graphs are edge-regular and, not surprisingly, the family of edge-regular graphs is much larger and more diverse than…

Consider the family of all finite graphs with maximum degree $\Delta(G)<d$ and matching number $\nu(G)<m$. In this paper we give a new proof to obtain the exact upper bound for the number of edges in such graphs and also characterize all…

We describe the structure of those graphs that have largest spectral radius in the class of all connected graphs with a given degree sequence. We show that in such a graph the degree sequence is non-increasing with respect to an ordering of…

The $k$-independence number of a graph $G$ is the maximum size of a set of vertices at pairwise distance greater than $k$. In this paper, for each positive integer $k$, we prove sharp upper bounds for the $k$-independence number in an…

We consider orbit partitions of groups of automorphisms for the symplectic graph and apply Godsil-McKay switching. As a result, we find four families of strongly regular graphs with the same parameters as the symplectic graphs, including…

In this paper the problem of the existence of regular nut graphs is addressed. A generalization of Fowler's Construction which is a local enlargement applied to a vertex in a graph is introduced to generate nut graphs of higher order. Let…

A tree $T$ in an edge-colored graph is a {\it proper tree} if no two adjacent edges of $T$ receive the same color. Let $G$ be a connected graph of order $n$ and $k$ be a fixed integer with $2\le k\le n$. For a vertex subset $S \subseteq…

We prove that the maximum degree $\Delta_n$ of a random series-parallel graph with $n$ vertices satisfies $\Delta_n/\log n \to c$ in probability, and $\mathbb{E}\, \Delta_n \sim c \log n$ for a computable constant $c>0$. The same result…